Gordan Savin

Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence

Using theta correspondence, we classify the irreducible representations of Mp 2 n in terms of the irreducible representations of SO 2 n +1 and determine many properties of this classification. This is a local Shimura correspondence which extends the well-known results of Waldspurger for n =1.

Functoriality and the inverse Galois problem

We prove that, for any primeand any even integer n, there are infinitely many exponents k for which PSp n (Fk) appears as a Galois group over Q. This generalizes a result of Wiese from 2006, which inspired this paper.

Motives with Galois group of type G2: An exceptional theta-correspondence

In this paper, we study an exceptional theta correspondence, obtained by restricting the minimal automorphic representation of the adjoint group of type E7 and rank 3 over Q to the dual pair GxPGSp6. Here G is the anisotropic form of G2 over Q; using the correspondence, we lift certain automorphic forms on G to holomorphic cusp forms on PGSp6. This lifting provides the first step in a project to construct motives of rank 7 and weight O over Q with Galois group of type G2.

Complementary series for hermitian quaternionic groups

Let G be a hermitian quaternionic group. We determine complementary series for representations of G induced from super-cuspidal representations of a Levi factor of the Siegel maximal parabolic subgroup of G.

Symplectic-orthogonal theta lifts of generic discrete series

0. Introduction. Let F be a non-Archimedean local field of characteristic zero. In this paper we study a correspondence between representations of symplectic groups Sp(n,F ) and special even-orthogonal split groups SO(2r,F ), where r ≥ 2. Letωn,r be the Weil representation of Sp(2nr,F ) attached to a nontrivial additive characterψF of F . We show that the correspondence arising by restricting the Weil representationωn,r to Sp(n,F )×SO(2r,F ) is functorial for generic square integrable representations. More precisely, let T be a smooth, irreducible representation of Sp(n,F ). Let 2(T, r) be the maximal T-isotypic quotient of ωn,r . The smallest r such that2(T, r) 6= 0 is called the first occurrence index of T. Now assume that T is a ψ-generic discrete series. (See (1.1) for the definition of ψ .) Let L(s,T) be the standard Lfunction attached to T as in [Sh1]. Then we have the following results. IfL(0,T)=∞, then the first occurrence index is n. Let τ ′ be an irreducible quotient of 2(T,n). Then τ ′ is a ψ ′-generic discrete series representation of SO(2n,F ), and for any discrete series representation δ of GL(m,F ) (m arbitrary), we have L(s,δ×T)= L(s,δ)L(s,δ×τ ′). If L(0,T) 6= ∞, then the first occurrence index is n+1. Then 2(T,n+1) has the unique irreducible ψ ′-generic quotient τ ′. Furthermore, τ ′ is a discrete series representation of SO(2n+2,F ), and for any discrete series representation δ of GL(m,F ) (m arbitrary), we have L(s,δ×τ ′)= L(s,δ)L(s,δ×T). We also have analogous results for ψ ′-generic discrete series of SO(2n,F ). We refer the reader to Section 2 for precise statements. Our results have a conjectural interpretation as follows. Consider inclusions of dual groups SO(2n,C)⊂ SO(2n+1,C)⊂ SO(2n+2,C). Let W ′(F ) be the Weil-Deligne group of F . The conjectural Langlands parameter of T is an admissible homomorphism (see [Bo]) φ :W ′(F )−→ SO(2n+1,C).

Exceptional Θ-Correspondences I

Let G be either a split SO(2n+2), or a split adjoint group of type En, (n=6,7,8), over a p-adic field. In this article we study correspondences arising by restricting the minimal representation of G to various dual pairs in G.

Representations of metaplectic groups II: Hecke algebra correspondences

Let k be a p-adic field with p odd. Let V + and V − be two quadratic spaces of dimension 2n+1, trivial discriminant, and trivial and non-trivial Hasse invariants, respectively. Then SO(V +) is a split, adjoint group of type Bn, while SO(V ) is its unique non-split inner form. Let S denote the category of smooth representations of SO(V ), and S 0 the component (in the sense of Bernstein [Be]) of S containing the trivial representation of SO(V ).

Modular forms on non-linear double covers of algebraic groups

We construct automorphic representations of non-linear two-fold covers of simply connected Chevalley groups via residues of Eisenstein series. In the process, we establish some basic results in representation theory of local groups.

Rank and matrix coefficients for simply laced groups

Abstract Let G be a simple, simply connected Chevalley group of type E over a local field k of characteristic 0. In this paper, we show that, amongst all non-trivial irreducible unitary representations of G, the matrix coefficients of the (unique) minimal representation of G have the slowest decay.

The smallest representations of nonlinear covers of odd orthogonal groups

We construct the smallest genuine representations of a nonlinear cover of the group